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Baltic Way
1999 Baltic Way
1
Going through the sums of a,b,c,d
Going through the sums of a,b,c,d
Source: Baltic Way 1999
December 23, 2010
algebra proposed
algebra
Problem Statement
Determine all real numbers
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
that satisfy the following equations
{
a
b
c
+
a
b
+
b
c
+
c
a
+
a
+
b
+
c
=
1
b
c
d
+
b
c
+
c
d
+
d
b
+
b
+
c
+
d
=
9
c
d
a
+
c
d
+
d
a
+
a
c
+
c
+
d
+
a
=
9
d
a
b
+
d
a
+
a
b
+
b
d
+
d
+
a
+
b
=
9
\begin{cases} abc + ab + bc + ca + a + b + c = 1\\ bcd + bc + cd + db + b + c + d = 9\\ cda + cd + da + ac + c + d + a = 9\\ dab + da + ab + bd + d + a + b = 9\end{cases}
⎩
⎨
⎧
ab
c
+
ab
+
b
c
+
c
a
+
a
+
b
+
c
=
1
b
c
d
+
b
c
+
c
d
+
d
b
+
b
+
c
+
d
=
9
c
d
a
+
c
d
+
d
a
+
a
c
+
c
+
d
+
a
=
9
d
ab
+
d
a
+
ab
+
b
d
+
d
+
a
+
b
=
9
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